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Geometry of Big Data – Tuesday session

All talks are summarised in my words which may not accurately represent the authors’ opinion. The focus is on aspects I found interesting. Please refer to the authors’ work for more details.

Session 1 – Graph-based persistence

The talk On the density of expected persistence diagrams and its kernel based estimation is given by Frederic Chazal. A draft is available on arxiv.

Grow circles around point data to generate a graph whenever other points meet the circle and produce a persistent homology of filtered simplicial complexes (e.g adding edges to possibly change homology). Persistent barcode and persistence diagrams encode the same information produced by this process.

Measures are nicer to work with than sets of points for statistical purposes. If the persistence diagram D is a random variable, then E[D] is a determnistic measure on R². Persistence images reveal E[D] and are more interpretable than persistance diagramms which may be too crowed for visual inspection with a large sample.

Persistence can be used as an additional feature on a dataset. For example, a random sample from the data set can be taken and the persistence diagram/image can be computed and compared between random samples giving us an idea of the stability of the homology.

Session 2 – Log-concave density estimation

The talk Log-concave density estimation: adaptation and high dimensions is given by Richard Samworth. The paper is available at Project Euclid.

To randomly sample a density f_0 there are generally two appraoches parametric and non-parametric methods. A density f is log-concave if log f is concave. The super level sets need to be convex. Univariate examples are normal, logistic and more. The class is closed under marginalisation, conditioning, convlution and linear transformations.

In an unbounded likelihood, the density surface is spiky. The log-concave density addresses this.

Session 3 – Infinite Width Neural Nets

The talk Infinite-Width Bounded-Norm Networks: A View from Function Space given by Nathan Srebro has two parts Infinite Width ReLU Nets and Geometry of Optimization Regularization and Inductive Bias.

Part 1: When we are learning we find a good fit (of weights) for the data. What kind of functions can be approxmiated by Neural Net? Essentially all, but the question is how large does the network have to be to approximate f to within error e. The question should be: what class function can be approximated by low norm Neural Nets? Another question should be: Given a bounded number of units what norm is required to approximate f to within any error e? The cost of the weights is taken as the parameter. This results in linear splines. A neural net with infinite width and one hidden layer solves the Green’s function.

Part 2: How does depth influence this? Deep learning should be considered with infinitive width and implemented with a finite approximation. Deep learning focuses on searching parameter space that maps into a richer function space.

Session 4

The talk Some geometric surprises in modern machine learningis given by Andrea Montanari.

Session 5

The talk Multi-target detection and cryo-EM imaging by autocorrelation analysis is given by Amit Singer.

Session 6

The talk Learning to Solve Inverse Problems in Imaging is given by Rebecca Willett.

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